Simplify the Radical Expression (6- square root of 3)/(2+ square root of 7)

$\frac{6 - \sqrt{3}}{2 + \sqrt{7}}$

Step 1

Multiply $\frac{6 - \sqrt{3}}{2 + \sqrt{7}}$ by $\frac{2 - \sqrt{7}}{2 - \sqrt{7}}$ .

$\frac{6 - \sqrt{3}}{2 + \sqrt{7}} \cdot \frac{2 - \sqrt{7}}{2 - \sqrt{7}}$

Step 2

Multiply $\frac{6 - \sqrt{3}}{2 + \sqrt{7}}$ by $\frac{2 - \sqrt{7}}{2 - \sqrt{7}}$ .

$\frac{(6 - \sqrt{3}) (2 - \sqrt{7})}{(2 + \sqrt{7}) (2 - \sqrt{7})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{(6 - \sqrt{3}) (2 - \sqrt{7})}{4 - 2 \sqrt{7} + \sqrt{7} \cdot 2 - {\sqrt{7}}^{2}}$

Step 4

Simplify.

$\frac{(6 - \sqrt{3}) (2 - \sqrt{7})}{- 3}$

Step 5

Expand $(6 - \sqrt{3}) (2 - \sqrt{7})$ using the FOIL Method.

Apply the distributive property.

$\frac{6 (2 - \sqrt{7}) - \sqrt{3} (2 - \sqrt{7})}{- 3}$

Apply the distributive property.

$\frac{6 \cdot 2 + 6 (- \sqrt{7}) - \sqrt{3} (2 - \sqrt{7})}{- 3}$

Apply the distributive property.

$\frac{6 \cdot 2 + 6 (- \sqrt{7}) - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{7})}{- 3}$

Step 6

Simplify each term.

Multiply $6$ by $2$ .

$\frac{12 + 6 (- \sqrt{7}) - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{7})}{- 3}$

Multiply $- 1$ by $6$ .

$\frac{12 - 6 \sqrt{7} - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{7})}{- 3}$

Multiply $2$ by $- 1$ .

$\frac{12 - 6 \sqrt{7} - 2 \sqrt{3} - \sqrt{3} (- \sqrt{7})}{- 3}$

Multiply $- \sqrt{3} (- \sqrt{7})$ .

Multiply $- 1$ by $- 1$ .

$\frac{12 - 6 \sqrt{7} - 2 \sqrt{3} + 1 \sqrt{3} \sqrt{7}}{- 3}$

Multiply $\sqrt{3}$ by $1$ .

$\frac{12 - 6 \sqrt{7} - 2 \sqrt{3} + \sqrt{3} \sqrt{7}}{- 3}$

Combine using the product rule for radicals.

$\frac{12 - 6 \sqrt{7} - 2 \sqrt{3} + \sqrt{3 \cdot 7}}{- 3}$

Multiply $3$ by $7$ .

$\frac{12 - 6 \sqrt{7} - 2 \sqrt{3} + \sqrt{21}}{- 3}$

Step 7

Move the negative in front of the fraction.

$- \frac{12 - 6 \sqrt{7} - 2 \sqrt{3} + \sqrt{21}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{12 - 6 \sqrt{7} - 2 \sqrt{3} + \sqrt{21}}{3}$

Decimal Form:

$0.91867792 \dots$

Simplify the Radical Expression (6- square root of 3)/(2+ square root of 7)

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